Generates the convex regression spline (called C-spline) basis matrix by integrating I-spline basis for a polynomial spline or the corresponding derivatives.

## Usage

```
cSpline(
x,
df = NULL,
knots = NULL,
degree = 3L,
intercept = TRUE,
Boundary.knots = NULL,
derivs = 0L,
scale = TRUE,
...
)
```

## Arguments

- x
The predictor variable. Missing values are allowed and will be returned as they are.

- df
Degree of freedom that equals to the column number of the returned matrix. One can specify

`df`

rather than`knots`

, then the function chooses`df - degree - as.integer(intercept)`

internal knots at suitable quantiles of`x`

ignoring missing values and those`x`

outside of the boundary. If internal knots are specified via`knots`

, the specified`df`

will be ignored.- knots
The internal breakpoints that define the splines. The default is

`NULL`

, which results in a basis for ordinary polynomial regression. Typical values are the mean or median for one knot, quantiles for more knots.- degree
The degree of C-spline defined to be the degree of the associated M-spline instead of actual polynomial degree. For example, C-spline basis of degree 2 is defined as the scaled double integral of associated M-spline basis of degree 2.

- intercept
If

`TRUE`

by default, all of the spline basis functions are returned. Notice that when using C-Spline for shape-restricted regression,`intercept = TRUE`

should be set even when an intercept term is considered additional to the spline basis in the model.- Boundary.knots
Boundary points at which to anchor the splines. By default, they are the range of

`x`

excluding`NA`

. If both`knots`

and`Boundary.knots`

are supplied, the basis parameters do not depend on`x`

. Data can extend beyond`Boundary.knots`

.- derivs
A nonnegative integer specifying the order of derivatives of C-splines. The default value is

`0L`

for C-spline basis functions.- scale
A logical value indicating if scaling C-splines is required. If

`TRUE`

by default, each C-spline basis is scaled to have unit height at right boundary knot. The corresponding I-spline and M-spline produced by`deriv`

methods will be scaled to the same extent.- ...
Optional arguments that are not used.

## Value

A numeric matrix of `length(x)`

rows and `df`

columns if

`df`

is specified or ```
length(knots) + degree +
as.integer(intercept)
```

columns if `knots`

are specified instead.
Attributes that correspond to the arguments specified are returned
mainly for other functions in this package.

## Details

It is an implementation of the closed-form C-spline basis derived from the recursion formula of I-splines and M-splines.

## References

Meyer, M. C. (2008). Inference using shape-restricted regression splines.
*The Annals of Applied Statistics*, 2(3), 1013--1033.

## Examples

```
library(splines2)
x <- seq.int(0, 1, 0.01)
knots <- c(0.3, 0.5, 0.6)
### when 'scale = TRUE' (by default)
csMat <- cSpline(x, knots = knots, degree = 2)
op <- par(mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0))
matplot(x, csMat, type = "l", ylab = "C-spline basis")
abline(v = knots, lty = 2, col = "gray")
isMat <- deriv(csMat)
msMat <- deriv(csMat, derivs = 2)
matplot(x, isMat, type = "l", ylab = "scaled I-spline basis")
matplot(x, msMat, type = "l", ylab = "scaled M-spline basis")
## reset to previous plotting settings
par(op)
### when 'scale = FALSE'
csMat <- cSpline(x, knots = knots, degree = 2, scale = FALSE)
## the corresponding I-splines and M-splines (with same arguments)
isMat <- iSpline(x, knots = knots, degree = 2)
msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)
## or using deriv methods (more efficient)
isMat1 <- deriv(csMat)
msMat1 <- deriv(csMat, derivs = 2)
## equivalent
stopifnot(all.equal(isMat, isMat1, check.attributes = FALSE))
stopifnot(all.equal(msMat, msMat1, check.attributes = FALSE))
```